Finite Group Databasebeta

← C3×#13313 #10631920

Subgroups — C3×#13313 #10631920

Order 768.

556 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D16C211yesmaximal, characteristic, complemented
2C2×C3×#472C212yesmaximal, complemented
3C2×C3×#361C212yesmaximal, complemented
4C3×#2113C211yesmaximal, characteristic, complemented
5C3×#1381C211yesmaximal, characteristic, complemented
6#13313C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×AGL(1,ℤ/8)—22no—
8C22×C3×D8C2212yes—
9C2×C3×#131C2211yescharacteristic
10C2×C3×D16—24no—
11C2×C3×D16—22no—
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14C3×#472—24no—
15C3×#472—24no—
16C2×C3×#108C2212yes—
17C2×C3×#100—22no—
18C3×#361—24no—
19C3×C22⋊C16—22no—
20C22×D16C2×C311yescharacteristic, complemented
21C2×#472C2×C312yescomplemented
22C2×#361C2×C312yescomplemented
23#2113C2×C311yescharacteristic, complemented
24#1381C2×C311yescharacteristic, complemented
25C2×C3×#41D411yescharacteristic
26C22×C3×D4—22no—
27C3×AGL(1,ℤ/8)—28no—
28C3×AGL(1,ℤ/8)—44no—
29C2×C3×SD16—42no—
30C2×C3×D8—22no—
31C2×C3×D8—22no—
32C2×C3×D8—22no—
33C2×C3×D8—22no—
34C2×C3×D8—22no—
35C2×C3×D8—42no—
36C2×C3×D8D414yes—
37C3×#131—22no—
38C3×#131—22no—
39C2×C3×M16—21no—
40C2×C3×M16—22no—
41C2×C3×M16D411yescharacteristic
42C22×C3×C8—21no—
43C22×C3×C8C2311yescharacteristic
44C3×D16—28no—
45C3×M32—22no—
46C2×C3×C16—22no—
47C2×C3×C16—21no—
48C2×C3×C16D412yes—
49C3×#108—22no—
50C3×#108D414yes—
51C3×#100—28no—
52C2×AGL(1,ℤ/8)—22no—
53C22×D8C22×C312yes—
54C2×#131C22×C311yescharacteristic
55C2×D16—24no—
56C2×D16—22no—
57C2×M32C22×C311yescharacteristic
58C22×C16C22×C311yescharacteristic
59#472—24no—
60#472—24no—
61C2×#108C22×C312yes—
62C2×#100—22no—
63#361—24no—
64C22⋊C16—22no—
65C24×C3—42no—
66C3×#41—22no—
67C3×#41—22no—
68C2×C3×Q8D811yescharacteristic
69C2×C3×D4—21no—
70C2×C3×D4—24no—
71C2×C3×D4—42no—
72C2×C3×D4—42no—
73C2×C3×D4—42no—
74C2×C3×D4—42no—
75C2×C3×D4—42no—
76C2×C3×D4D811yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4C2×D411yescharacteristic
79C3×SD16—48no—
80C3×D8—24no—
81C3×D8—24no—
82C3×D8—48no—
83C3×D8—42no—
84C3×D8—42no—
85C3×M16—22no—
86C3×M16—24no—
87C3×M16—42no—
88C3×M16—42no—
89C2×C3×C8—21no—
90C2×C3×C8—42no—
91C2×C3×C8—41no—
92C2×C3×C8—41no—
93C2×C3×C8C2×D412yes—
94C2×C3×C8C2×D411yescharacteristic
95C2×C3×C8C2×D412yes—
96C2×C3×C8#4111yescharacteristic
97C3×C16—22no—
98C3×C16—41no—
99C2×#41C3×D411yescharacteristic
100C22×D4—22no—
101AGL(1,ℤ/8)—28no—
102AGL(1,ℤ/8)—44no—
103C2×SD16—42no—
104C2×D8—22no—
105C2×D8—22no—
106C2×D8—22no—
107C2×D8—22no—
108C2×D8—22no—
109C2×D8—42no—
110C2×D8C3×D414yes—
111#131—22no—
112#131—22no—
113C2×M16—21no—
114C2×M16—22no—
115C2×M16C3×D411yescharacteristic
116C22×C8—21no—
117C22×C8C23×C311yescharacteristic
118D16—28no—
119M32—22no—
120C2×C16—22no—
121C2×C16—21no—
122C2×C16C3×D412yes—
123#108—22no—
124#108C3×D414yes—
125#100—28no—
126C23×C3—21no—
127C23×C3—44no—
128C23×C3—82no—
129C23×C3—82no—
130C23×C3—164no—
131C23×C3#13511yescharacteristic
132C3×Q8—22no—
133C3×D4—22no—
134C3×D4—42no—
135C3×D4—44no—
136C3×D4—44no—
137C3×D4—82no—
138C3×D4—82no—
139C2×C3×C4—21no—
140C2×C3×C4—41no—
141C2×C3×C4—41no—
142C2×C3×C4#12012yes—
143C2×C3×C4#12111yescharacteristic
144C2×C3×C4C2×D811yescharacteristic
145C2×C3×C4AGL(1,ℤ/8)12yes—
146C3×C8—41no—
147C3×C8—44no—
148C3×C8#12011yescharacteristic
149C3×C8#12112yes—
150C3×C8#12711yescharacteristic
151C24—42no—
152#41—22no—
153#41—22no—
154C2×Q8C3×D811yescharacteristic
155C2×D4—21no—
156C2×D4—24no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4C3×D811yescharacteristic
163C22×C4—21no—
164C22×C4C2×C3×D411yescharacteristic
165SD16—48no—
166D8—24no—
167D8—24no—
168D8—48no—
169D8—42no—
170D8—42no—
171M16—22no—
172M16—24no—
173M16—42no—
174M16—42no—
175C2×C8—21no—
176C2×C8—42no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8C2×C3×D412yes—
180C2×C8C2×C3×D411yescharacteristic
181C2×C8C2×C3×D412yes—
182C2×C8C3×#4111yescharacteristic
183C16—22no—
184C16—41no—
185C22×C3—22no—
186C22×C3—42no—
187C22×C3—41no—
188C22×C3—84no—
189C22×C3—164no—
190C22×C3—164no—
191C22×C3—164no—
192C22×C3#44912yes—
193C22×C3#46611yescharacteristic
194C3×C4—41no—
195C3×C4#39211yescharacteristic
196C3×C4#44712yes—
197C3×C4#46911yescharacteristic
198C23—21no—
199C23—44no—
200C23—82no—
201C23—82no—
202C23—164no—
203C23C3×#13511yescharacteristic
204Q8—22no—
205D4—22no—
206D4—42no—
207D4—44no—
208D4—44no—
209D4—82no—
210D4—82no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4C3×#12012yes—
215C2×C4C3×#12111yescharacteristic
216C2×C4C2×C3×D811yescharacteristic
217C2×C4C3×AGL(1,ℤ/8)12yes—
218C8—41no—
219C8—44no—
220C8C3×#12011yescharacteristic
221C8C3×#12112yes—
222C8C3×#12711yescharacteristic
223C2×C3—22no—
224C2×C3—81no—
225C2×C3—164no—
226C2×C3#199911yescharacteristic
227C2×C3#222112yes—
228C22—22no—
229C22—41no—
230C22—42no—
231C22—84no—
232C22—164no—
233C22—164no—
234C22—164no—
235C22C3×#44912yes—
236C22C3×#46611yescharacteristic
237C4—41no—
238C4C3×#39211yescharacteristic
239C4C3×#44712yes—
240C4C3×#46911yescharacteristic
241C3#1331311yescharacteristic, direct product factor, Sylow
242C2—22no—
243C2—81no—
244C2—164no—
245C2C3×#199911yescharacteristic
246C2C3×#222112yes—
subgroupG/H
center, socleC22×C3#466
commutator, Frattini, cosocleC22×C8C23×C3
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×C3×D16C2semidirect
2. C2×C3×#472C2semidirect
3. C2×C3×#361C2semidirect
4. C3×#2113C2semidirect
5. C3×#1381C2semidirect
6. #13313C3direct (N × H)
20. C22×D16C2×C3semidirect
21. C2×#472C2×C3semidirect
22. C2×#361C2×C3semidirect
23. #2113C2×C3semidirect
24. #1381C2×C3semidirect
241. C3#13313direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×D162
C2×C3×#4722
C2×C3×#3612
C3×#21132
C3×#13812
#133133